# Performance

There are several metrics for helicopter performance.  Top speed, range and acceleration are probably familiar to car drivers, but other factors like hover ceiling (how high can a helicopter hover) and loiter time (how long can a helicopter stay in the air without landing) are more specific to helicopters and less familiar to the average person.  Below we will discuss important metrics for helicopter performance and then discuss some technical aspects.

## Helicopter Performance Metrics

### Hover Ceiling

The hover ceiling is the maximum (pressure) altitude at which a helicopter can hover.  At higher altitudes air is less dense which makes both the rotor and engine less efficient – more power is required by the rotor and the engine is not capable of producing as much power.  With speed, a helicopter can fly above its hover ceiling - the rotor requires less power in cruise than hover, contrary to what you might guess.

A chart like the one below may be used to specify the hover ceiling.  Remember, air density is the important value here.  However, since pressure altitude and temperature are more readily available (and together they mostly determine the density) they are used instead of density.  Since hover ceiling decreases with weight, charts like the one below use weight on the horizontal axis (pressure altitude on the vertical axis).  You will notice separate lines in the chart for each temperature - higher temperatures correspond to lower air density and therefore a lower pressure altitude ceiling.

There is another consideration here – ground effect.  When a helicopter is near the ground the rotor is more efficient and can hover at lower air density (higher pressure attitude).  For this reason, two charts like the one above are normally provided: one for in ground effect (IGE) and another for out of ground effect (OGE).  A little more about the physics of this is provided below.

Hovering requires more power than cruising.  This makes hover ceilings quite important:  it’s possible to cruise at a given altitude, but not be able to slow and hover there.  Why?  Aircraft maintain altitude (counteract gravity) by accelerating air downward.  In hover, a helicopter is engulfed in this downward flowing column of air.  This makes the main rotor less efficient, like a kayak going upstream.  With forward speed the helicopter moves out of this column of air, partially accessing less disturbed air in front of it.  (The speed of this downward flowing air may be estimated using momentum theory as described here.)

The column of air thrown down by the helicopter must slow to have 0 vertical velocity at ground level.  If the helicopter is near the ground, this increases the air pressure under the helicopter and reduces the vertical speed of air through the rotor.  This is the ground effect phenomena mentioned above, which effectively makes the rotor more efficient.  This becomes relevant when the main rotor is within a diameter of the ground, and increases as the rotor gets closer to the ground (most prevalent when the helicopter is resting on the ground).

### Loiter Time

A metric particularly important for smaller helicopters used in law enforcement, news and sightseeing is loiter time.  This is the maximum amount of time the helicopter can stay aloft before refueling.  This requires the helicopter to fly at a specific airspeed, often around 60kts, called loiter speed.  Loiter speed is the airspeed with lowest fuel flow rate - smaller or larger speeds will burn fuel faster and reduce time aloft.   An example plot below shows fuel flow rate versus airspeed, marking the loiter airspeed.  Knowing this fuel flow rate $$R$$ and the aircraft fuel capacity $$C$$, one can estimate the loiter time as $$C/R$$.  In reality, $$R$$ decreases as weight decreases (i.e. with less fuel) and hence this underestimates loiter time (assuming $$R$$ was based on carrying max fuel).

### Range

A helicopter’s range is the distance it can fly without refueling (assuming no wind).  Range is dependent on the weight (passengers and cargo onboard), airspeed and air density.  Of course, range shrinks with added weight.  The airspeed which provides maximum range is not the same as loiter speed.  Flying faster, while burning more fuel per unit time, burns less fuel per unit distance.   Hence, range increases as airspeed increases above loiter speed.  This is true up to a limit called the max range airspeed (MRA).  Beyond MRA, the fuel burn per unit distance increases.  Flying at MRA will get the helicopter from point A to B while consuming the least fuel.  However, helicopters are often flown faster for passenger convenience.

## Breaking Down the Power Required

The total power required by the helicopter may be subdivided into profile power, parasite power, induced power, and others we’ll categorize as miscellaneous.   The plot below gives a rough idea of how these values change with airspeed.  We’ll explain all these values below.

### Profile Power

Profile power is the power required to overcome drag on the rotor blades.  This is often about 20% of the power required in hover.  At high speeds this grows quickly - first due to the larger airspeed experienced by the advancing rotor blade and at higher speeds due to retreating blade stall and advancing blade compressibility effects.

We'll provide a simple calculation here.  We'll consider a section of the rotor $$dr$$ at distance $$r$$ from the hub, and assume it maintains a constant drag coefficient $$c$$ around the azimuth.  A single such section (one blade) encounters a drag force $$D=\rho c v_\psi ^2dr/2$$, where $$v_\psi$$ is the airspeed of the section (we've lumped the product of the chord length and drag coefficient together into $$c$$).  The airspeed will be $$\Omega r$$ due to the rotor rotating at speed $$\Omega$$, plus the speed of the helicopter $$v$$ in the reference frame of the blade section, which is $$v \sin\psi$$.  We'll use the convention where $$\psi$$ is the azimuth angle of the blade and $$\psi =0$$ corresponds to a blade over the tail of the helicopter.  Averaging the associated power $$Dv_\psi$$ over the azimuth gives $$$$P = \frac{1}{2\pi} \int_0^{2\pi}d\psi Dv_\psi = \frac{1}{2\pi} \int_0^{2\pi} d\psi \rho c(\Omega r+v\sin \psi )^3dr/2 \label{eq:profpwr1}$$.$$

From here the arithmetic is tedious.  One way to solve this is expand the cubed sum to get a sum of terms with powers of $$\sin^n \psi$$.  Anti-derivatives of such terms can be looked up and then evaluated at $$2\pi$$ and $$0$$ to give the following $$$$P = dr \rho c \Omega^3 r^3 (1 + \frac{3v^2}{2 \Omega^2 r^2} ) / 2 \label{eq:prof2}$$.$$

To compute profile power for the entire rotor, Equation \eqref{eq:prof2} can be multiplied by the number of blades and integrated from the blade root to $$r=R$$.  However, Equation \eqref{eq:prof2} is enough to see the trends.  Profile power is clearly minimized at hover and increases with the square of the helicopter airspeed $$v$$.  Recall that this was assuming a constant drag coefficient.  At large speeds $$v$$ portions of a blade suffer from compressibility effects when advancing ($$\psi \approx 90^o$$) and stall when retreating ($$\psi \approx 270^o$$), causing $$c$$ and therefore $$P$$ to increase further.

### Parasite Power

Parasite power is the power required to pull the helicopter (minus the rotors) through the air.  This is associated with drag on the fuselage, skids and tail surfaces.  At low speeds this is negligible, but this drag increases with the square of the airspeed so that the power increases with the cube of the airspeed.

A simple method used in many calculations is to provide a table of fuselage drag, normalized by dynamic pressure $$\rho v^2/2$$.  These values may be provided in a 2D table, as a function of fuselage angle of attack $$\alpha$$ and sideslip $$\beta$$: $$c=c(\alpha ,\beta )$$.  Since the fuselage drag is then $$D=\rho v^2 c/2$$, the parasite power becomes $$P=Dv=\rho v^3 c/2$$.

### Induced Power

Induced power is what we eluded to in the hover performance section above.   The velocity of air thrown down by the main rotor is called induced velocity (its magnitude can be estimated using momentum theory).  It causes the net direction of air movement relative to a main rotor blade to be tilted down from the horizontal.  The diagram below shows a cross section of a rotor blade and the velocity of air relative to it: the left blue arrow labeled V (total).  Lift is perpendicular to this velocity and hence it’s not directly upward, but it's partially tilted rightward in the diagram.  The horizontal component of that lift vector (dashed line near the top of the diagram) is called induced drag and pushes back against the rotor rotation.  The power required to overcome this component of lift (keep the rotor turning at full speed) is induced power.  As you might expect from earlier discussion, this power is largest at hover and low speed and reduces at higher speeds where the rotor accesses “cleaner” air (with less induced velocity).

### Miscellaneous Power

Miscellaneous power includes losses in the drive system, power required for avionics and hydraulic pumps, the tail rotor, etc.  This is normally less than 20% of the total power required.